The Symbolism of the Yuunicorn
The choice to base Yuu's magical girl form and doppel off of a unicorn is an incredibly solid one-- not just for the visual, but from a folkloric perspective as well. The first thing that jumps out at me is that unicorns have long been associated with (or emblematic of) purity and virtue, which fits perfectly with Yuu's obsession with doing "good things" to make up for being a "bad person"-- similar to how the unicorn in the tapestry "The Unicorn Purifies Water" uses its horn to purify a stream for the animals of the forest, Yuu does what she can to help others and get rid of "bad things". This extends to her negative self-perception, as well; she's more than willing to get rid of herself if doing so would make Shii feel even a little bit better. (Shii actively avoids telling Yuu that she has, in fact, seen Yuu in her nightmares because of this.)
In fact, unicorns are so pure that their horns were said to be able to relieve wounds and instantly cleanse poison (Images et connaissance de la licorne: (Fin du Moyen Âge - xixe siècle), Bruno Faidutti, 1996), and according to Claudius Aelianus, drinking from one could supposedly shield a person from all toxins and diseases (Attributs et symboles dans l'art profane: dictionnaire d'un langage perdu (1450-1600), Guy de Tervarent, 1997). Yuu serves a similar function to Shii: Not only does she kill "bad guys" to try to keep Shii safe, she looks out for her in mundane ways as well, earning money to support the two of them, handling chores, and being right there by Shii's side to calm her down when she's had an especially bad nightmare.
Another piece of unicorn horn-related trivia that I think is cool is this description of the unicorn (aka. monoceros) from the Cosmas Indicopleustes, Christian Topography, book 11:
They speak of him as a terrible beast and quite invincible, and say that all his strength lies in his horn. When he finds himself pursued by many hunters and on the point of being caught, he springs up to the top of some precipice whence he throws himself down and in the descent turns a somersault so that the horn sustains all the shock of the fall, and he escapes unhurt.
Sounds familiar, both to how resilient she is and the fact that she did indeed once throw herself from a cliff and survive (though she wasn't entirely happy about that...).
The unicorn's gentleness is good here, too, I think. Portraying Yuu as a creature renowned for how sweet, docile, and protective it is captures her personality quite nicely, and adds to the feeling of dissonance between her nature and the horrific acts she commits. Similarly, assigning a creature that's ontologically "good" to a character who sees herself as irreparably "bad" makes for a very fun contrast.
But something really interesting to me that I haven't seen anyone talk about yet is how well Yuu being a unicorn ties in with the "virgin capture" myths.
According to medieval legends, one seeking to hunt a unicorn must lure it out of hiding with a virgin girl. The unicorn is attracted by her scent (hmm...) and/or aura of purity, and loves her so much that it will come to rest its head on her chest or lap, at which point a knight or hunter can strike it down. This excerpt from Vestigia (vol.3, issue 1, by Angelica Federici) is rather on the nose:
His character, indomitable by nature, could be pacified only thanks to a virgin, who had the power to lull the creature into a profound sleep. It is almost impossible to overlook her duplicitous nature, as object of purity and betrayer of the unicorn.
Both Shii and Kei could be seen as the "virgin" to Yuu's unicorn, pulling her in but ultimately leading to her death (literal death in Shii's case; death of the mind/self and eventually literal death in Kei's case). It's not an intentional betrayal, but the theme of the unicorn being helplessly drawn to a girl out of love only to meet its demise is the same. Kei even sees herself as two-faced, and apologizes for as much in a flashback in Beachside Bonds:
Kei: How do you see me? Do I look kind to you? Chaste? Shy? It's all a lie. I'm sorry... All of it, from start to finish. All lies.
The virgin's ability to lull the unicorn to rest is fun if you view the virgin as Kei, too, since the botched double suicide was what led Yuu to become the Sleepwalking Ghost.
And whether you take the virgin as Shii or Kei, this section from the same article feels relevant:
Although the virgin is witnessing the brutal slaying of the unicorn, she seems passive and detached. [...] In courtly love circles where the legend gained momentum, it is interesting to note that the lover unicorn dies at the hands of the virgin in the quest for love, her position nevertheless is still described as fixed and immutable (Beer 1986: iv). She is a passive bystander, a chess pawn in the Hortus Conclusus, without any hope of redemption.
Both Shii and Kei view themselves as awful, irredeemable people, though they take different approaches to how they handle that. Kei saw herself as too stagnant and broken to live, which is part of why she brought up that suicide pact with Yuu, whereas Shii figures that she's too far gone to be saved anyway; she may as well keep making bad decisions.
Also also: Some more contemporary versions of the unicorn myth say that they can only be tamed by people who are kind, innocent, or otherwise pure of character (rather than simply being a virgin in the most literal sense), which I think is fitting with Yuu's choice to dedicate herself to helping Shii. Those are typically stories where the maiden befriends the unicorn, rather than ones where she's being used as a lure, but that still tracks. While Shii doesn't see herself as good, Yuu certainly believes she is, and Shii is the only living person that Yuu is fully obedient to.
Much like how the unicorn isn't usually a violent creature (its flightiness is part of what makes it so impossible to capture), Yuu is happy to help people in whatever way she can-- but when it comes to killing bad guys, she cannot be reined in or swayed by anyone other than the figurative maiden who tamed her.
With all of that said: I'm sure that some of the things I've brought up here are simply lucky coincidences, rather than intentional reasons why the unicorn was chosen as one of Yuu's motifs, haha. Either way, though, I've gotta reiterate that I think this was an absolutely perfect decision for her character.